1. Field of the Invention
The present invention relates to a method of forming, from data obtained by exploration of a zone of a heterogeneous medium, a model representative of the distribution in the zone of a physical quantity (at least partly) free of the presence of correlated noises that may be contained in the data.
The method applies, for example, to the quantification of the acoustic impedance in an underground zone.
2. Description of the Prior Art
The process of seeking a model that adjusts to experimental measurements has been developed in nearly all the fields of the sciences or technology. Such an approach is known under various names: least-squares method for parameters estimation, for inverse problem solution. For a good presentation of this approach within the context of geosciences, one may for example refer to:
Tarantola, A.: “Inverse Problem Theory: Method for Data Fitting and Model Parameter Estimation”, Elsevier, Amsterdam, 1987.
It can be noted that the term “least squares” refers to the square of the norm in the data space for quantifying the difference between the response of a model (which is the image of the model by a previously selected modelling operator) and the data using, a cost function that has to be minimized to solve the problem. Using the square of the norm to define the cost function is just a practical convenience, and it is not fundamentally essential. Besides, many authors use, for various reasons, another definition of the cost function but this definition remains based on the use of the norm, or of a semi-norm, in the data space. Finally, considerable latitude exists for selecting the norm (or the semi-norm) in the data space (the use of the Euclidean norm is not required). In the case of noise-containing data, the solution can substantially depend on the choice made at this stage. More developments concerning this problem, are referred to in the following publications:
Tarantola, A.: “Inverse Problem Theory: Method for Data Fitting and Model Parameter Estimation”, Elsevier, Amsterdam, 1987; Renard and Lailly, 2001; Scales and Gersztenkorn, 1988 ; Al-Chalabu, 1992.
The measuring results often contain errors. Modelling noises add further to these measuring errors when the experimentation is compared with modelling results: modellings are never perfect and therefore always correspond to a simplified view of reality. The noise will therefore be described hereafter as the superposition of:
non correlated components (white noise for example),
correlated components, which means that the existence of a noise on a measuring sample translates into the existence of a noise of equal nature on certain neighbouring measuring points; modelling noises typically belong to this category.
When the data contain correlated noises, the quality of the model estimated by solving the inverse problem can be seriously affected thereby. As already mentioned, no modelling operator is perfect. It is therefore the work of the whole community of the people involved in the identification of parameters describing a model which is hindered by the existence of correlated noises. Among these people, seismic exploration practitioners are among the most concerned ones: in fact, their data have a poor or even very bad signal-to-noise ratio. This is the reason why correlated noise filtering techniques are an important part of seismic data processing softwares. The most conventional techniques use a transform (Fourier transform for example) where signal and noise are located in different areas of space, thus allowing separation of the signal from the noise. A general presentation of the conventional methods for noise elimination on seismic data is discussed in the book by Yilmaz:
Yilmaz, O. 1987: “Seismic Data Processing”, Investigation in Geophysics No.2, Society of Exploration Geophysicists, Tulsa, 1987.
However, these techniques are not perfect: they presuppose that a transformation allowing complete separation of the signal and of the noise has been found. In this context, correlated noises are particularly bothersome because they can be difficult to separate from the signal (which is also correlated) and can have high amplitudes. It is therefore often difficult to manage the following compromise: either the signal is preserved, but a large noise residue remains, or the noise is eliminated, but then the signal is distorted. These filtering techniques can be implemented before solving the inverse problem: they constitute then a preprocessing applied to the data. The quality of the inverse problem solution then widely depends on the ability of the filters to eliminate the noise without distorting the signal.
This approach is introduced by Nemeth et al. in the following publication:
Nemeth T. et al. (1999), “Least-Square Migration of Incomplete Reflection Data”, Geophysics, 64, 208–221 which constitutes an important advance for the inversion of data disturbed by high-amplitude correlated noises. The authors propose eliminating the noise by solving an inverse problem: after defining the space of the correlated noises as the image space of a vector space B (space of the noise-generating functions) by a linear operator T and the signal as the image space of a vector space M (models space) by a modelling operator F (assumed to be linear), they seek the signal in F(M) and the noise in T(B) whose sum is as close as possible (in the sense of the Euclidean norm or, on a continuous basis, of the norm L2) to the measured datum. This technique constitutes an important advance insofar as it allows elimination (or at least very substantial reduction) of high-amplitude correlated noises (surface waves for example) that are difficult to separate from the signal by means of conventional filtering techniques. However, according to the authors, an increase by an order of magnitude of the computing time required for solution of the conventional inverse problem, that is without seeking the correlated component of the noise, is the price to pay for these performances. Furthermore, the result obtained with the method is extremely sensitive to any inaccuracy introduced upon definition of operator T: this is the ineluctable compensation for the high aptitude of the method to discriminate signal and noise.